Algebra & Numerical Reasoning
Rate Problems
Applying formulas for Speed (S=V×T) and Work (W=P×T).
Problems appear in government logistics, staffing allocation, and transport/travel scenarios.
What the exam tests
Missing Variable Calculations
Given two of three variables, solve for the third using Speed (S = V × T) or Work (W = P × T).
Relative Speed Scenarios
Objects moving in relation to an environment, such as a boat going upstream vs. downstream where the river's flow adjusts effective speed.
Proportional Workload/Staffing
Determine how many employees are needed to finish a task in a specific timeframe, assuming constant work rate per person.
Efficiency and Consumption Rates
Multi-step problems with rates like MPG. These often require finding total gallons used before solving for distance.
Unit Rate Identification
Find a specific per-unit value, such as job applications per opening or items purchased per dollar.
Scale and Mapping
Determine real-world distances from a map scale (e.g., ¼ cm = 20 km).
Formula Mastery and Algebraic Rearrangement
Key rules
- ›Speed: S = V × T. Rearrange to V = S ÷ T or T = S ÷ V depending on the missing variable.
- ›Work: W = P × T. Rearrange to P = W ÷ T or T = W ÷ P.
- ›Identify which part of the scenario maps to Work/Distance, Power/Velocity, and Time before plugging in numbers.
Common traps
- !Staffing problems are inverse: doubling the number of workers halves the time. Do not treat it as a direct proportion.
Combined and Relative Rates
Key rules
- ›Downstream speed = boat speed + current speed.
- ›Upstream speed = boat speed − current speed.
- ›For MPG problems, calculate total gallons first, then use the rate formula to find distance or time.
Common traps
- !Relative speed problems require identifying the direction of travel before deciding to add or subtract the environmental rate.
Unit Consistency, Data Filtering, and Estimation
Key rules
- ›Convert all variables to compatible units (hours, miles, kilograms) before applying any formula.
- ›Eliminate irrelevant information such as names, dates, and unrelated figures. Extract only the rate variables.
- ›Use mental estimation to verify the answer is logically reasonable for the scenario described.
Common traps
- !Mixed units (e.g., minutes vs. hours) cause off-by-60 errors. Always normalize time units before calculating.
Try one
Jill drove across a 0.3-mile bridge in 20 seconds. What was her average speed in miles per hour?
Convert 20 seconds to hours: 20 ÷ 3,600 = 1/180 hour. Speed = 0.3 ÷ (1/180) = 0.3 × 180 = 54 mph.
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